STT 2820 - Normal Distribution (Ch 5)

Front

What is a normal distribution?

Back

A normal distribution is a continuous probability distribution characterized by a symmetric bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values further away from the mean taper off equally in both directions.

Front

What are the mean and standard deviation in a normal distribution?

Back

The mean is the average of all data points, and the standard deviation measures the dispersion or spread of the data points around the mean.

Front

What is the empirical rule (68-95-99.7 rule)?

Back

The empirical rule states that for a normal distribution: about 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and about 99.7% falls within three standard deviations.

Front

How do you calculate the z-score?

Back

The z-score is calculated using the formula: z = (X - μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.

Front

What does a z-score represent?

Back

A z-score represents the number of standard deviations a data point is from the mean. A positive z-score indicates the value is above the mean, while a negative z-score indicates it is below the mean.

Front

What is the significance of the area under the normal curve?

Back

The area under the normal curve represents the total probability of all outcomes, which equals 1. Specific areas can represent probabilities for certain ranges of values.

Front

How do you find the proportion of data below a certain value in a normal distribution?

Back

To find the proportion of data below a certain value, calculate the z-score for that value and then use the standard normal distribution table to find the corresponding probability.

Front

What is the cutoff for the middle 50% of a normal distribution?

Back

The cutoff for the middle 50% is found by determining the first and third quartiles (Q1 and Q3), which correspond to the 25th and 75th percentiles.

Front

What is the role of the mean in determining the shape of a normal distribution?

Back

The mean determines the center of the distribution. A higher mean shifts the curve to the right, while a lower mean shifts it to the left.

Front

What is the role of the standard deviation in determining the shape of a normal distribution?

Back

The standard deviation determines the width of the distribution. A larger standard deviation results in a wider and flatter curve, while a smaller standard deviation results in a narrower and taller curve.